"the hypercube graph. It is one of 5 Platonic graphs, each a skeleton of its Platonic solid. An extension is the three dimensional ""k""-ary Hamming graph, which for ""k"" = 2 is the cube graph. Graphs of this sort occur in the theory of parallel processing in computers. Miscellaneous cubes Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube"
"which can be simplified by noting that 5/10 = 3/6 = 1/2: formula_75 The square root can also be expressed by a periodic continued fraction, but the above form converges more quickly with the proper ""x"" and ""y"". The cube root of two (2 or ≈ 1.259921...): (A) ""Standard notation"" of ""x"" = 1, ""y"" = 1, and 2""z - y"" = 3: formula_76 (B) Rapid convergence with ""x"" = 5, ""y"" = 3 and 2""z - y"" = 253: formula_77 Pogson's ratio (100 or ≈ 2.511886...), with ""x"" = 5, ""y"" = 75 and 2""z - y"" = 6325:"
"Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"" are The cube root operation is not distributive with addition or subtraction. In"
"nested cube roots such as can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation If we solve this equation, we find that ""x"" = 2. More generally, we find that is the positive real root of the equation ""x"" − ""x"" − ""n"" = 0 for all ""n"" > 0. For ""n"" = 1, this root is the plastic number ""ρ"", approximately equal to 1.3247. The same procedure also works to get as the real root of the equation ""x"" + ""x"" − ""n"" = 0 for"
"root is coprime with 10 are somewhat easier since the digits are scrambled in consistent ways, as we shall see in the next section. An easy task for the beginner is extracting cube roots from the cubes of 2 digit numbers. For example, given 74088, determine what two digit number, when multiplied by itself once and then multiplied by the number again, yields 74088. One who knows the method will quickly know the answer is 42, as 42 = 74088. Before learning the procedure, it is required that the performer memorize the cubes of the numbers 1-10: Observe that there"
"number of sides). This is because the sum of the angles of any ""n""-gon is 180° × (""n"" – 2) and so the measure of each angle of any regular ""n""-gon is 180° × (""n"" – 2) ÷ ""n"". Thus for example the entry ""45°: square"" means that, with ""n"" = 4, 180° ÷ ""n"" = 45°, and the number of degrees in each angle of a square is (""n"" – 2) × 45° = 90°. For cube roots of non-real numbers that appear in this table, one has to take the principal value, that is the cube root with"
"the cube root of numbers having many digits in the ""Aryabhatiya"" (section 2.5). Cube root In mathematics, a cube root of a number ""x"" is a number ""y"" such that ""y"" = ""x"". All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8, denoted , is 2, because 2 = 8, while the other cube roots of 8 are −1 + ""i"" and −1 − ""i"". The three cube roots of −27""i"""
"""square super-root"", or ""super square root"" has two equivalent notations, formula_128 and formula_129. It is the inverse of formula_130 and can be represented with the Lambert W function: The function also illustrates the reflective nature of the root and logarithm functions as the equation below only holds true when formula_132: Like square roots, the square super-root of ""x"" may not have a single solution. Unlike square roots, determining the number of square super-roots of ""x"" may be difficult. In general, if formula_134, then ""x"" has two positive square super-roots between 0 and 1; and if formula_135, then ""x"" has one"
"Square root of 5 The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as: It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are: which can be rounded down to 2.236 to within 99.99% accuracy. The approximation (≈ 2.23611) for the square root"
"Cube (algebra) In arithmetic and algebra, the cube of a number is its third power: the result of the number multiplied by itself twice: It is also the number multiplied by its square: This is also the volume formula for a geometric cube with sides of length , giving rise to the name. The inverse operation of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. Both cube and cube root are odd functions: The cube"
"for ""irrational number"" being translated into Latin as ""surdus"" (meaning ""deaf"" or ""mute""). Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to ""unresolved irrational roots"". A square root of a number ""x"" is a number ""r"" which, when squared, becomes ""x"": Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign: Since the square of every"
"some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the ""principal cube root"", denoted with the radical sign . The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers: for example, the cube of any cube root of 8 is 8, but the three cube roots of 8 are 8, −4 + 4""i"", and −4 − 4""i"". The cube roots"
"real number is a positive real number, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5""i"" and −5""i"", where ""i"" represents a number whose square is . A cube root of a number ""x"" is a number ""r"" whose cube is ""x"": Every real number ""x"" has exactly one real cube root, written formula_4. For example, Every real number has two additional complex cube roots. Expressing the degree of an ""n""th root in its exponent form, as in formula_37, makes"
"solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the complex cube root of a complex number. Quartic equations can also be solved in terms of cube roots and square roots. The calculation of cube roots can be to traced back to Babylonian mathematicians from as early as 1800 BCE. In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube;"
"the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the ""casus irreducibilis"", in which"
"this problem, and in 1837 Pierre Wantzel proved it to be impossible because the cube root of 2 is not a constructible number. The cube has three uniform colorings, named by the colors of the square faces around each vertex: 111, 112, 123. The cube has three classes of symmetry, which can be represented by vertex-transitive coloring the faces. The highest octahedral symmetry O has all the faces the same color. The dihedral symmetry D comes from the cube being a prism, with all four sides being the same color. The lowest symmetry D is also a prismatic symmetry, with"
"or the other of the two complex cube roots of 1. For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula If we write ""x"" as where ""r"" is a non-negative real number and ""θ"" lies in the range then the principal complex cube root is This means that in polar coordinates, we are taking the cube root of the radius and dividing the"
"equation in sum to zero. If a cubic is plotted in the Cartesian plane, the real root can be seen graphically as the horizontal intercept of the curve. But further, if the complex conjugate roots are written as then is the abscissa (the positive or negative horizontal distance from the origin) of the tangency point of a line that is tangent to the cubic curve and intersects the horizontal axis at the same place as does the cubic curve; and is the square root of the tangent of the angle between this line and the horizontal axis. With one real"
"only if formula_125, with formula_126 being the maximum power of 2 which divides formula_127. According to Dickson's ""History Of Numbers"", the following formula of Cipolla will find square roots of powers of prime modula: Taking the example in the wiki article we can see that this formula above does indeed take square roots of prime power modula. As Now solve for formula_134 via: Now create the formula_136 and formula_137 (See for mathematica code showing this above computation, remembering that something close to complex modular arithmetic is going on here) As such: and the final equation is: Cipolla's algorithm In computational"
"this question in ""Scientific American"", Kay R. Pechenick DeVicci and several other readers showed that the answer for the (3,4) case is the square root of the smaller of two real roots of the polynomial formula_7, which works out to approximately 1.007435. For formula_8, the optimal side length of the largest square in an formula_5-dimensional hypercube is either formula_10 or formula_11, depending on whether formula_5 is even or odd respectively. Prince Rupert's cube In geometry, Prince Rupert's cube (named after Prince Rupert of the Rhine) is the largest cube that can pass through a hole cut through a unit cube,"
"group: Find the square root of 152.2756. Find the cube root of 4192 to the nearest hundredth. The principal ""n""th root of a positive number can be computed using logarithms. Starting from the equation that defines ""r"" as an ""n""th root of ""x"", namely formula_80 with ""x"" positive and therefore its principal root ""r"" also positive, one takes logarithms of both sides (any base of the logarithm will do) to obtain The root ""r"" is recovered from this by taking the antilog: For the case in which ""x"" is negative and ""n"" is odd, there is one real root ""r"""
"square root of five can be written as a rational linear combination of roots of unity: The square root of 5 appears in various identities discovered by Srinivasa Ramanujan involving continued fractions. For example, this case of the Rogers–Ramanujan continued fraction: Square root of 5 The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can"
"Square root of 2 The square root of 2, or the (1/2)th power of 2, written in mathematics as or , is the positive algebraic number that, when multiplied by itself, gives the number 2. Technically, it is called the principal square root of 2, to distinguish it from the negative number with the same property. Geometrically the square root of 2 is the length of a diagonal across a square with sides of one unit of length; this follows from the Pythagorean theorem. It was probably the first number known to be irrational. As a good rational approximation for"
"is a pattern in the rightmost digit: adding and subtracting with 1 or 3. Starting from zero: There are two steps to extracting the cube root from the cube of a two digit number. Say you are asked to extract the cube root of 29791. Begin by determining the one's place (units) of the two digit number. You know it must be one, since the cube ends in 1, as seen above. Note that every digit corresponds to itself except for 2, 3, 7 and 8, which are just subtracted from ten to obtain the corresponding digit. The second step"
"as the square super-root, terminology for other super roots can be based on the normal roots: ""cube super-roots"" can be expressed as formula_138; the ""4th super-root"" can be expressed as formula_139; and the """"n"" super-root"" is formula_124. Note that formula_124 may not be uniquely defined, because there may be more than one ""n"" root. For example, ""x"" has a single (real) super-root if ""n"" is ""odd"", and up to two if ""n"" is ""even"". Just as with the extension of tetration to infinite heights, the super-root can be extended to formula_142, being well-defined if 1/""e"" ≤ ""x"" ≤ ""e"". Note"
"is: find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √−15 and 5 − √−15. Cardano called this ""sophistic,"" because he saw no physical meaning to it, but boldly wrote ""nevertheless we will operate"" and formally calculated that their product does indeed equal 40. Cardano then says that this answer is “as subtle as it is useless”. It is a common misconception that Cardano introduced complex numbers in solving cubic equations. Since (in modern notation) Cardano's formula for a root of the polynomial ""x"" + ""px"" + ""q"""
"and , since, in each case, we have the sum of a cube root of () with a cube root () and moreover the product of these two roots is, in each case, equal to . Note that and that . Thus Cardano's formula, written unambiguously to give the three roots, is where the cube roots expressed as radicals are defined to be any pair of cube roots whose product is . If and are real and , this is the same thing as requiring that the cube roots be complex conjugates, while if and are real and , the"
"formula_57 is a Legendre symbol. From formula_58 follows that formula_59. Thus ""c"" is a quadratic residue modulo ""p"". Hence formula_60 and therefore The relation formula_55 is not a requirement because square roots modulo other primes can be computed too. E.g., Rabin proposes to find the square roots modulo primes by using a special case of Berlekamp's algorithm. Decoding produces three false results in addition to the correct one, so that the correct result must be guessed. This is the major disadvantage of the Rabin cryptosystem and one of the factors which have prevented it from finding widespread practical use. If"
"this required the construction, now known to be impossible, of the length . A method for extracting cube roots appears in ""The Nine Chapters on the Mathematical Art"", a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. The Greek mathematician Hero of Alexandria devised a method for calculating cube roots in the 1st century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes. In 499 CE Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave a method for finding"
"cube root of 29791 is 31. Another example: This process can be extended to find cube roots that are 3 digits long, by using arithmetic modulo 11. These types of tricks can be used in any root where the order of the root is coprime with 10; thus it fails to work in square root, since the power, 2, divides into 10. 3 does not divide 10, thus cube roots work. To approximate a common log (to at least one decimal point accuracy), a few log rules, and the memorization of a few logs is required. One must know: From"
"is not zero, no two of these roots are equal. Since the quadratic term of this cubic polynomial is zero, the roots are related by the equation The linear and constant coefficients (""g"" and ""g"", respectively) are related to the roots by the equations (see Elementary symmetric polynomial). The roots ""e"", ""e"", and ""e"" of the equation formula_52 depend on and can be expressed in terms of theta functions. As before, let, then Since formula_59 and formula_60, then these can also be expressed as theta functions. In simplified form, Where formula_46 is the Dedekind eta function. In the case of"
"however, easy to show that most numbers are not perfect cubes because ""all"" perfect cubes must have digital root 1, 8 or 9. That is their values modulo 9 may be only −1, 1 and 0. Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by 3: Every positive integer can be written as the sum of nine (or fewer) positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes: The equation has"
"has three roots, and can be written alternatively as: which can be expanded as: formula_52 is a volume of an unstable state between formula_33 and formula_34. The cubic equations are identical. Therefore, from the linear terms in these equations, formula_55 can be solved: From the quadratic terms, ""B"" can be solved: And from the cubic terms, ""C"" can be solved: Since formula_33, formula_34 and formula_31 were tabulated for many fluids with formula_32 as a parameter, it is a simple matter to compute ""B"" and ""C"" in the saturation region of these fluids. The results are generally in agreement with those"